Base 24

From Wikipedia, the free encyclopedia

Numeral systems by culture
Hindu-Arabic numerals
Indian
Eastern Arabic
Khmer		 Indian family
Brahmi
Thai
East Asian numerals
Chinese
Suzhou
Counting rods		 Japanese
Korean 
Alphabetic numerals
Abjad
Armenian
Cyrillic
Ge'ez		 Hebrew
Greek (Ionian)
Āryabhaṭa
 
Other systems
Attic
Babylonian
Egyptian
Etruscan		 Mayan
Roman
Urnfield
List of numeral system topics
Positional systems by base
Decimal (10)
2, 4, 8, 16, 32, 64
1, 3, 9, 12, 20, 24, 30, 36, 60, more…
v • d • e

The base-24 system is a numeral system with 24 as its base.

There are 24 hours in a day, so our time keeping system includes a base-24 component. See also base 12.

                           Decimal Equivalent
       10  twenty four      24             24
      100  ?               24^2 =         576
     1000  ?               24^3 =      13 824
   10 000  ?               24^4 =     331 776
  100 000  ?               24^5 =   7 972 624
1 000 000  ?               24^6 = 191 102 976

The digits used for numerals ten (10) to twenty three (23) may be the letters "A" through to "P" ("I" and "O" are skipped to prevent confusion with the digits 1 and 0 in some typefaces).

[edit] Fractions

Quadrovigesimal fractions are usually either very simple

1/2 = 0.C
1/3 = 0.8
1/4 = 0.6
1/6 = 0.4
1/8 = 0.3
1/9 = 0.2G
1/C = 0.2
1/G = 0.1C
1/J = 0.18

or complicated

1/5  = 0.4K4K4K4K... recurring (easily rounded to 0.5 or 0.4K)
1/7  = 0.3A6LDH3A6... recurring
1/A  = 0.29E9E9E9... recurring (rounded to 0.2A)
1/B  = 0.248HAMKF6D248.. recurring (rounded to 0.24)
1/D  = 0.1L795CN3GEJB1L7.. recurring (rounded to 0.1L)
1/P  = 0.11111... recurring (rounded to 0.11)
1/11 = 0.0P0P0P... recurring (rounded to 0.0P) (1/(5*5))

As explained in recurring decimals, whenever a fraction is written in "decimal" notation, in any base, the fraction can be expressed exactly (terminates) if and only if all the prime factors of its denominator are also prime factors of the base. Thus, in base-10 (= 2×5) system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: ¹⁄8 = ¹⁄(2*2*2), ¹⁄20 = ¹⁄(2×2×5), and ¹⁄500 (2²×5³) can be expressed exactly as 0.125, 0.05, and 0.002 respectively. ¹⁄3 and ¹⁄7, however, recur (0.333... and 0.142857142857...). In the duodecimal (= 2×2×3) system, ¹⁄8 is exact; ¹⁄20 and ¹⁄500 recur because they include 5 as a factor; ¹⁄3 is exact; and ¹⁄7 recurs, just as it does in base 10.

In practical applications, the nuisance of recurring decimals is encountered less often when quadrovigesimal (or duodecimal) notation is used.

However when recurring fractions do occur in quadrovigesimal notation, they sometimes have a very short period when they are numbers containing one or two factors of five, as 5² = 25 is adjacent to 24. The other adjacent number, 23, is a prime number. So powers of five look funny in the quadrovigesimal notation:

51 =          5
52 =         11
53 =         55
54 =        121
55 =        5A5
56 =       1331
57 =       5FF5
58 =      14641

The multiples of decimal hundred are 44, 88, CC, GG, LL, 110, etc.

[edit] Natural languages

Umbu-Ungu,[1] also known as Kakoli, is reported to have base-24 numerals[2]. Tokapu means 24, tokapu talu means 24×2 = 48, and tokapu tokapu means 24×24 = 576.

[edit] References

  1. ^ Gordon, Raymond G., Jr., ed. (2005), “Umbu-Ungu”, Ethnologue: Languages of the World (15 ed.), <http://www.ethnologue.com/show_language.asp?code=ubu>. Retrieved on 16 March 2008 
  2. ^ Bowers, Nancy & Lepi, Pundia (1975), “Kaugel Valley systems of reckoning”, Journal of the Polynesian Society 84 (3): 309-324, <http://www.ethnomath.org/resources/bowers-lepi1975.pdf> 
Languages